Einstein summation convention

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11: 34.3 Basic Properties: $\mathit{3}j$ Symbol

… ►In the summations (34.3.16)–(34.3.18) the summation variables range over all values that satisfy the conditions given in (34.2.1)–(34.2.3). Similar conventions apply to all subsequent summations in this chapter. …

12: William P. Reinhardt

… ►He has recently carried out research on non-linear dynamics of Bose–Einstein condensates that served to motivate his interest in elliptic functions. …

13: Bibliography G

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Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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External Links:

ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

Encodings:

BibTeX

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On the computation of generalized Fermi-Dirac and Bose-Einstein integrals. Comput. Phys. Comm. 74 (2), pp. 233–238.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§25.12(iii), §25.18(i), 5th item.

Encodings:

BibTeX

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Multilateral summation theorems for ordinary and basic hypergeometric series in $\mathrm{U}(n)$ . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.

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External Links:

ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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14: Bibliography B

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Integral representations and summations of the modified Struve function. Acta Math. Hungar. 141 (3), pp. 254–281.

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External Links:

ISSN 0236-5294, Document, FullText, MathReview (Eugenia N. Petropoulou), ZentralBlatt

Referenced by:

§11.7(v), §11.7(v).

Encodings:

BibTeX

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Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.

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External Links:

Document, MathReview (T. M. Apostol), ZentralBlatt

Referenced by:

§24.16(ii).

Encodings:

BibTeX

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Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§24.16(iii), §24.8(ii).

Encodings:

BibTeX

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Stability of repulsive Bose-Einstein condensates in a periodic potential. Phys. Rev. E (3) 63 (036612), pp. 1–11.

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External Links:

Document

Referenced by:

§29.19(i).

Encodings:

BibTeX

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The Borel transform and its use in the summation of asymptotic expansions. Stud. Appl. Math. 105 (2), pp. 83–113.

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External Links:

ISSN 0022-2526, Document, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

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15: 18.20 Hahn Class: Explicit Representations

… ►Here we use as convention for (16.2.1) with $b_q=-N$, $a_1=-n$, and $n=0,1,\mathrm{\dots },N$ that the summation on the right-hand side ends at $k=n$. …

16: 18.26 Wilson Class: Continued

… ►Here we use as convention for (16.2.1) with $b_q=-N$, $a_1=-n$, and $n=0,1,\mathrm{\dots },N$ that the summation on the right-hand side ends at $k=n$. …

17: 4.12 Generalized Logarithms and Exponentials

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4.12.9 $$\psi (x)=\mathrm{\ell }+\underset{\mathrm{\ell }\text{ times}}{\underbrace{\ln \mathrm{\cdots }\ln }}x,$$ $x>1$,

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Symbols:

$\ln z$: principal branch of logarithm function, $x$: real variable, $\psi (x)$: generalized logarithm and $\mathrm{\ell }$: positive integer

Permalink:

http://dlmf.nist.gov/4.12.E9

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.24):

The $\mathrm{\ell }$-times repeated application of $\ln$, previously given as ${\ln }^{(\mathrm{\ell })}$, has been rewritten using a more conventional notation.

See also:

Annotations for §4.12 and Ch.4

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4.12.10

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Defines:

$\mathrm{\ell }$: positive integer (locally)

Symbols:

$\ln z$: principal branch of logarithm function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.12.E10

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.24):

The $\mathrm{\ell }$-times repeated application of $\ln$, previously given as ${\ln }^{(\mathrm{\ell })}$, has been rewritten using a more conventional notation.

See also:

Annotations for §4.12 and Ch.4

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18: 5.19 Mathematical Applications

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§5.19(i) Summation of Rational Functions

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19: 17.18 Methods of Computation

… ►The two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. …

20: 34.10 Zeros

… ►Similarly the $\mathit{6}j$ symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four $\mathit{3}j$ symbols in the summation. …